Question

The determinant $\left|\begin{array}{ccc}{y}^2& -xy& x^2\\ a& b& c\\ a^{\prime }& b^{\prime }& c^{\prime }\end{array}\right|$ is equal to

• A. $\left|\begin{array}{cc}bx+ay& cx+by\\ b^{\prime }x+a^{\prime }y& c^{\prime }x+b^{\prime }y\end{array}\right|$
• B. $\left|\begin{array}{cc}ax+by& bx+cy\\ a^{\prime }x+b^{\prime }y& b^{\prime }x+c^{\prime }y\end{array}\right|$
• C. $\left|\begin{array}{cc}bx+cy& ax+by\\ b^{\prime }x+c^{\prime }y& a^{\prime }x+b^{\prime }y\end{array}\right|$
• D. none of these

Answer 1

The correct option is B $\left|\begin{array}{cc}ax+by& bx+cy\\ a^{\prime }x+b^{\prime }y& b^{\prime }x+c^{\prime }y\end{array}\right|$

Given determinant $\left|\begin{array}{ccc}{y}^2& -xy& x^2\\ a& b& c\\ a^{\prime }& b^{\prime }& c^{\prime }\end{array}\right|$
$=y^2(b{c}^{\prime }-b^{\prime }c)+xy(a{c}^{\prime }-a^{\prime }c)+x^2(a{b}^{\prime }-a^{\prime }b)$
Taking option A,$\left|\begin{array}{cc}bx+ay& cx+by\\ b^{\prime }x+a^{\prime }y& c^{\prime }x+b^{\prime }y\end{array}\right|$
$=(bx+ay)(c^{\prime }x+b^{\prime }y)-(cx+by)(b^{\prime }x+a^{\prime }y)$
$=b{c}^{\prime }{x}^2+b{b}^{\prime }xy+a{c}^{\prime }xy+a{b}^{\prime }{y}^2-c{b}^{\prime }{x}^2-c{a}^{\prime }xy-b{b}^{\prime }xy-a^{\prime }b{y}^2$
$=x^2(b{c}^{\prime }-b^{\prime }c)+xy(a{c}^{\prime }-a^{\prime }c)+y^2(a{b}^{\prime }-a^{\prime }b)$
Value of option A determinant does not matches with the given determinant.
Taking option B, $\left|\begin{array}{cc}ax+by& bx+cy\\ a^{\prime }x+b^{\prime }y& b^{\prime }x+c^{\prime }y\end{array}\right|$
$=(ax+by)(b^{\prime }x+c^{\prime }y)-(bx+cy)(a^{\prime }x+b^{\prime }y)$
$=a{b}^{\prime }{x}^2+a{c}^{\prime }xy+b{b}^{\prime }xy+b{c}^{\prime }{y}^2-b{a}^{\prime }{x}^2-b{b}^{\prime }xy-a^{\prime }cxy-b^{\prime }c{y}^2$
$=y^2(b{c}^{\prime }-b^{\prime }c)+xy(a{c}^{\prime }-a^{\prime }c)+x^2(a{b}^{\prime }-a^{\prime }b)$
Value of option B determinant matches with the given determinant.

Value of option C determinant does not matches with the given determinant. It give negative value.